There is a single idea underlying much of the confusion in discussions of personal identity: the belief in a soul.
I use this term for a quality or substance which resides in a person throughout his life and is somehow responsible for his identity, and which (here is the problem) is not captured by a complete description of the person's physical and psychological state. Often, it is a hidden assumption.
That's a nice definition of the soul, quite similar to the provable properties
of the "first person", once we will define it precisely (in the Thaetetus way). And comp will
entails, *as a theorem*, the existence of the soul, then!
The comp reason why the soul or the first person is never captured by any
complete third person description is akin to the reason truth and knowledge are
not arithmetizable (as opposed to provable and consistent).
Godel's theorem & Co. makes universal machine a highly non trivial type of being.